Paper · Oct 13, 2025
The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor
As on arXiv
Abstract
In this article, we give a new proof of a result due to J. Kim, which states that the Ricci tensor of a gradient Ricci soliton with dimension and harmonic Weyl tensor has at most three distinct eigenvalues. This result constitutes an essential step in the classification of such manifolds, originally established by J. Kim in dimension and subsequently extended to dimensions . Our proof offers two notable advantages: it is shorter and does not require the use of any specialized moving frame.
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.1Ricci Solitons
Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.
- 12B.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.